With one and two piece mattresses that have fiber or are "baffled" you need a device to remove the water. One low cost option is the Venturi pump . These are acceptable for full motion mattresses although relatively slow.

They are acceptable for mattresses with a single fiber layer, but are not recommended for mattresses with several layers of fiber. Fiber mattresses need to be pumped until they are flat. Anything less will allow the fiber to shift during moving and the mattress will probably never be the same again.